Polytope of Type {12,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*864o
if this polytope has a name.
Group : SmallGroup(864,4669)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 36, 216, 36
Order of s0s1s2 : 12
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {12,12,2} of size 1728
Vertex Figure Of :
   {2,12,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
   4-fold quotients : {6,12}*216c
   9-fold quotients : {4,12}*96c
   12-fold quotients : {6,4}*72
   18-fold quotients : {4,6}*48c
   36-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,12}*1728ab
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 2.
      24 facets:
         12 of {12}*24
         12 of {6}*12
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 3.
      12 facets:
         12 of {12}*24
      12 vertex figures:
         12 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      24 facets:
         18 of {4}*8
         6 of {12}*24
      12 vertex figures:
         12 of {12}*24

Permutation Representation (GAP) : 
s0 := ( 1, 3)( 2, 4)( 5,11)( 6,12)( 7, 9)( 8,10)(13,27)(14,28)(15,25)(16,26)(17,35)(18,36)(19,33)(20,34)(21,31)(22,32)(23,29)(24,30);;
s1 := ( 1, 5)( 2, 7)( 3, 6)( 4, 8)(10,11)(13,17)(14,19)(15,18)(16,20)(22,23)(25,29)(26,31)(27,30)(28,32)(34,35);;
s2 := ( 2, 4)( 5,13)( 6,16)( 7,15)( 8,14)( 9,25)(10,28)(11,27)(12,26)(18,20)(21,29)(22,32)(23,31)(24,30)(34,36);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) : 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(36)!( 1, 3)( 2, 4)( 5,11)( 6,12)( 7, 9)( 8,10)(13,27)(14,28)(15,25)(16,26)(17,35)(18,36)(19,33)(20,34)(21,31)(22,32)(23,29)(24,30);
s1 := Sym(36)!( 1, 5)( 2, 7)( 3, 6)( 4, 8)(10,11)(13,17)(14,19)(15,18)(16,20)(22,23)(25,29)(26,31)(27,30)(28,32)(34,35);
s2 := Sym(36)!( 2, 4)( 5,13)( 6,16)( 7,15)( 8,14)( 9,25)(10,28)(11,27)(12,26)(18,20)(21,29)(22,32)(23,31)(24,30)(34,36);
poly := sub<Sym(36)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) : 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
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